multi-objective optimization of peel and shear strengths

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Mathematical Biosciences and Engineering, x(x): xxx–xxxDOI:Received:Accepted:Published:

Type of article

Multi-Objective Optimization of Peel and Shear Strengths in UltrasonicMetal Welding Using Machine Learning-Based Response SurfaceMethodology

Yuquan Meng1, Manjunath Rajagopal1, Gowtham Kuntumalla1, Ricardo Toro1, HanyangZhao1, Ho Chan Chang1, Sreenath Sundar1, Srinivasa Salapaka1, Nenad Miljkovic1, PlacidFerreira1, Sanjiv Sinha1, Chenhui Shao1,∗

1 Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign,Urbana, IL 61801, USA

* Correspondence: Email: [email protected]; Tel: +1-217-300-4750.

Abstract: Ultrasonic metal welding (UMW) is a solid-state joining technique with varied industrialapplications. Despite of its numerous advantages, UMW has a relative narrow operating window andis sensitive to variations in process conditions. As such, it is imperative to quantitatively characterizethe influence of welding parameters on the resulting joint quality. The quantification model can besubsequently used to optimize the parameters. Conventional response surface methodology (RSM)usually employs linear or polynomial models, which may not be able to capture the intricate, nonlin-ear input-output relationships in UMW. Furthermore, some UMW applications call for simultaneousoptimization of multiple quality indices such as peel strength, shear strength, electrical conductivity,and thermal conductivity. To address these challenges, this paper develops a machine learning (ML)-based RSM to model the input-output relationships in UMW and jointly optimize two quality indices,namely, peel and shear strengths. The performance of various ML methods including spline regression,Gaussian process regression (GPR), support vector regression (SVR), and conventional polynomial re-gression models with different orders is compared. A case study using experimental data shows thatGPR with radial basis function (RBF) kernel and SVR with RBF kernel achieve the best predictionaccuracy. The obtained response surface models are then used to optimize a compound joint strengthindicator that is defined as the average of normalized shear and peel strengths. In addition, the casestudy reveals different patterns in the response surfaces of shear and peel strengths, which has not beensystematically studied in the literature. While developed for the UMW application, the method can beextended to other manufacturing processes.

Keywords: ultrasonic metal welding, mechanical strength, response surface methodology, processoptimization, machine learning

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1. Introduction

UMW is a solid-state joining method with various industrial applications including lithium-ion bat-tery assembly [1, 2, 3], automotive body construction [4, 5, 6], and electronic packaging [7, 8, 9].Among the advantages of UMW over conventional fusion welding techniques are the ability to joindissimilar metals, reduced energy consumption, and short welding cycles [1, 2, 10]. In addition, UMWis more environmentally friendly and free of solidification defects. It is thus considered as a promisingjoining technology contributing to the emerging trends in advanced manufacturing including elec-trification and lightweighting. It is documented that UMW can accommodate junctions of variouscombinations, including metal/metal, polymer/polymer, and metal/polymer [11, 12]. A typical UMWsystem is comprised of generator, transducer, booster, horn, and anvil, as shown by Fig. 1. These com-ponents convert high-frequency voltage to high-frequency mechanical vibration, which subsequentlyjoins workpieces together.

1

controller

hornbooster

anvil

transducer

Clamping Force

Anvil(stationary)

Battery tabs

Horn

Ultrasonic metal welding machine20 kHz vibration

Figure 1. Schematic of a UMW system.

Despite of its numerous advantages, UMW has a relative narrow operating window and is sen-sitive to variations in process conditions. As such, quality control of UMW has received extensiveattention, especially in industrial applications. Existing research on predicting UMW quality usuallyemploys two types of approaches, namely, finite element simulations (e.g., [13, 14]) and data-drivenonline process monitoring (e.g., [15, 16, 17, 18]). Xi et al. [13] developed finite element models topredict the mechanical performance of UMW joints under lap shear tests based on measured effectivethickness and bonding length, which are obtained from microscopic analysis. Shen et al. [14] utilizedfinite element methods to predict the temperature profile and microstructure evolution during ultrasonicwelding. Nonetheless, the finite element models are inherently computationally expensive, prohibitingcost-effective process optimization in practice and they fail to establish an end-to-end connection be-tween input parameters and the output quality. Online process monitoring methods use in-situ sensingsignals, e.g., power, vibration displacement, and audible signals, to infer on the joint quality. Keysteps in the development of these methods include [17] (i) sensor selection, (ii) signal preprocessing,(iii) feature extraction, (iv) feature selection, e.g., [15], and (iv) quality prediction using artificial in-telligence approaches, e.g., [16]. However, such methods assume that process parameters have beenoptimized offline and cannot account for the impact of process parameters on output quality.

The joint quality of UMW is influenced by a set of welding parameters including welding am-plitude, welding time, clamping pressure, and welding pressure. Identifying the best combination ofparameters, namely, process optimization, is still an open yet challenging question. In some indus-

Mathematical Biosciences and Engineering Volume x, Issue x, xxx–xxx

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tries, trial and error approaches are still commonly used. Such methods require a large number ofwelding experiments, which is costly, time-consuming, and less accurate. Furthermore, it has beenestablished that UMW has strong process variability due to uncontrollable process conditions such assurface contamination [3, 19] and tool wear [17, 20, 21, 22, 23, 24]. Although the adverse effects ofsuch conditions have been qualitatively characterized in the literature, the direct link between thesefactors and joint quality is still missing. Therefore, a more effective and efficient approach for responsesurface modeling and process optimization is critically needed.

RSM is a data-driven methodology that explores the relationships between process inputs and one ormultiple process outputs [25]. Statistical models such as linear and polynomial models are commonlyused in RSM to characterize the input-output relationships based on experimental data obtained froma sequence of designed experiments. RSM is particularly useful when there is a lack of understandingon the physical process. In manufacturing, RSM has been successfully applied to different processesincluding machining [26, 27], friction spot welding [28], arc welding [29], resistance spot welding[30], and UMW [1]. A third-order polynomial function (or cubic function) was developed in [1]to characterize the influence of welding pressure and welding time on the resulting peel strength.The process complexity in UMW may lead to strong nonlinearities in the input-output relationshipsthat cannot be adequately characterized by polynomial models. This motivates the application of MLmodels for response surface modeling due to their superior modeling capabilities. Some recent studieshave explored using ML models for other manufacturing processes. For instance, an RBF was usedto model the relationships between process parameters and performance responses in an electricaldischarge machining process [31]. Yet, no studies exist on ML-based RSM for UMW.

The joint quality of UMW is often characterized by mechanical strength, which is commonly eval-uated by peel and shear tests. While both testing methods are widely used in existing studies, thedifferences in the response surfaces of peel and shear strengths are not well understood. Furthermore,some applications call for the optimization of quality indices in addition to joint strength, such as elec-trical conductivity in electric vehicle batteries [2] and thermal conductivity in heat exchangers [32].Consequently, a multi-objective optimization approach is highly desirable.

In this paper, we employ different ML models including spline regression, GPR, and SVR to char-acterize the influence of two welding parameters, namely, welding time and welding pressure, on theshear and peel strengths. Different kernels are used in GPR and SVR. We also compare the model-ing performance of ML methods and polynomial models with first, second, and third orders basedon a cross-validation procedure. Afterwards, the differences in response surfaces for shear and peelstrengths are analyzed and discussed. Finally, the shear and peel strengths are jointly optimized usingthe best-performing response surface models.

This remainder of this paper is organized as follows. In Section 2, we introduce the response surfacemodeling methods used in this research. Section 3 presents the details of the experimental design andmechanical testing procedure. The main results are presented and discussed in Section 4. Section 5concludes the paper and discusses possible future research.

2. Machine learning-based response surface methodology

In this section, we present a brief introduction to polynomial regression and ML models used,including spline regression, GPR, and SVR.


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2.1. Polynomial regression

One important goal of RSM is to construct a function f (x) that can estimate the input-output re-lation of a process. Polynomial regression has been popularly used in conventional RSM. Supposex1, x2, ..., xn is the training input, and y1, y2, ..., yn is the corresponding output. n is the number of train-ing samples. The parameters in a polynomial model can be obtained by minimizing the predictionerror, which is given by the Eq. (2.1).

L =

n∑i=1

(yi − yi)2 =

n∑i=1

(yi − P(xi))2, (2.1)

where P(x) is the polynomial function to be found [33].Polynomial functions with lower orders may not be able to express the complex response surfaces.

On the other hand, a higher-order model generally has better modeling capability, but it may sufferfrom an overfitting issue. The choice of orders can be made based on the practitioner’s experience orusing a more rigorous model selection method. Furthermore, polynomial regression can capture theglobal tendency of the whole training dataset but may lose some delicate local topography.

2.2. Spline regression

Spline regression utilizes piecewise polynomial functions, called spline functions, to describe acomplex relationship or function. Spline regression has a certain order of smoothness, i.e., a splinefunction f , defined on [x0, xk], that satisfies:

f (x) =

P0(x), x0 ≤ x < x1

P1(x), x1 ≤ x < x2

...

Pk−1(x), xk−1 ≤ x ≤ xk

(2.2)

and

f (p)(x) ∈ C(x0, xk), p = 1, 2, ...m − 1, (2.3)

where P0(x), P1(x), ..., Pk−1(x) are polynomials of order m. Cubic functions, i.e., m = 3, is a commonchoice. The points x0, x1, ..., xk are called knots. In multiple input variable problems, the spline functionis constructed as the tensor product of each piecewise polynomial Pi(x).

Spline regression models are capable of capturing the localized variations while ensuring high-levelsmoothness. In our case, there are two input variables. Hence the 2D spline function is employed.We use tensor product of each piecewise polynomial Pi(x), displayed by Eq. (2.4). In this paper,the piecewise cubic function and the tensor product are used for 2D spline to construct the 2D splinefunction [34].

f (x, y) = Pi(x)P j(y), xi ≤ x < xi+1, y j ≤ y < y j+1. (2.4)

2.3. GPR

A Gaussian process is a collection of random variables such that the joint distribution of every finitesubset of random variables is multivariate Gaussian. It can be considered as a Gaussian distribution


5

over functions, i.e.,

f (x) ∼ GP(µ(x),K(x, x′) + δi jσ2), (2.5)

where m(x) and K(x, x′) are the mean and covariance functions. δi j is the Kronecker delta function. σ2

is the variance of the noise. The prediction of GP is made by modeling the joint distribution based ontraining dataset, also called observations. Specifically, the collection of training points and test pointsare jointly multivariate Gaussian distributed [35]:[

ff ∗

]∼ N

([µ

µ∗

],

[K(X, X) + σ2I K(X, X∗)

K(X∗, X) K(X∗, X∗)

]). (2.6)

Then the prediction is given by:

f ∗|X, y, X∗ ∼ N( f ∗,Σ∗), (2.7)f ∗ = µ∗ + K(X∗, X)[K(X, X) + σ2I]−1(y − µ), (2.8)Σ∗ = K(X∗, X∗) + K(X∗, X)[K(X, X) + σ2I]−1K(X, X∗). (2.9)

The performance of GPR is largely determined by the kernel function. Common choices for kernelfunctions include RBF and rational quadratic (RQ) function, which are shown by Eq. (2.10) and Eq.(2.11), respectively. In this paper, RBF and RQ kernels are used.

K(x, y) = exp(−‖x − y‖2

2σ20

). (2.10)

K(x, y) =(1 + ‖x − y‖2

)−α. (2.11)

2.4. SVR

The basic idea of SVR is to find a function f (x) that can estimate the targets yi given training dataxi and has at most ε deviation while ensuring the flatness of f (x) [36]. The simplest form for f (x) isthe linear function, i.e.,

f (x) =< w, x > +b, (2.12)

where w is the weight, b represents bias, and < · , · > refers to the inner product. Then the problem canbe written as

min12‖w‖2 + C

n∑i

ξi,

subject to ‖yi− < w, xi > −b‖ ≤ ε + ξi,

ξi ≥ 0,

(2.13)

where C is the trade-off between flatness of f and the deviation.Similar to GPR, the concept of kernels is introduced in SVR to solve nonlinear problems. The most

commonly adopted kernels are:


6

• polynomial kernel:

K(x, y) = (< x, y > +1)d. (2.14)

• RBF kernel:

K(x, y) = exp(−‖x − y‖2

2σ20

). (2.15)

By solving the Lagrangian dual [36], the SVR for nonlinear problem can be described in the fol-lowing optimization problem:

min−12

n∑i, j=1

(αi − α∗i )(α j − α

∗j)K(xi, x j) − ε

n∑i=1

(αi + α∗i ) +

n∑i=1

yi(αi − α∗i )

subject ton∑

i=1

(αi − α∗i ) = 0 and αi, α

∗i ∈ [0,C].

(2.16)

and f has the form of

f (x) =

n∑i=1

(αi − α∗i )K(x, xi) + b. (2.17)

3. Joining configurations and mechanical testing

3.1. Experimental setup

0.254 mm thick 110-copper sheet is used for the welding experiments. The copper sheet is trimmedinto small pieces in the dimension of 76 mm × 51 mm. Before welding, cleaning wipes are used toremove the pollutants on the copper specimen. The Branson Ultraweld L20 Spot Welder, shown inFig 1(b), is employed to perform UMW experiment. Time mode is selected in this experiment.

In UMW, welding amplitude, welding time, clamping pressure, and welding pressure are dominantfactors influencing the joint strength. Welding amplitude refers to the amplitude of the horn’s cyclicmovement. It does not only directly affects the material softening process but also influences the rateof frictional heat generation. Welding time determines the duration of a welding process. It can changethe final joint quality through influencing the amount of frictional heat generated. Clamping pressure(or trigger pressure) refers to the pre-load pressure that triggers ultrasonics. Welding pressure refersto the pressure applied vertically to the workpieces during the welding process. In this study, weldingamplitude, ranging from 40 µm to 55 µm, and welding time, ranging from 0.40 s to 0.80 s, are selectedto be the process/independent variables (inputs), whereas clamping and welding pressure are fixed at50 psi. The parameters and number of repeats for each testing configuration are shown in Table 1.

3.2. Mechanical testing

Two types of mechanical testing, i.e., peel and shear testing, are applied to measure the jointstrength. Fig. 2(a) and (b) shows the configurations of shear and peel tests, respectively. In a shear


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Table 1. Welding configurations and replicatesWelding Time (s) Welding Amplitude (µm) Number of Repeats-Shear Number of Repeats-Peel

0.40 40 10 40.45 40 10 40.50 40 11 40.55 40 10 40.60 40 10 40.65 40 5 40.70 40 5 40.75 40 5 40.80 40 5 40.40 45 10 40.45 45 10 40.50 45 10 40.55 45 9 40.60 45 10 40.65 45 5 40.70 45 5 40.75 45 5 40.80 45 5 40.40 50 9 40.45 50 10 40.50 50 13 40.55 50 9 40.60 50 10 40.65 50 5 40.70 50 5 40.75 50 5 40.80 50 5 40.40 55 10 40.45 55 10 40.50 55 10 40.55 55 10 40.60 55 10 40.70 55 5 40.80 55 5 4

test, the load is prescribed parallel to the bonding surface. Whereas in peel test, the load is appliedperpendicular to the joint surface. In both tests, the instantaneous load is recorded until the two layersare completely separated. Fig. 2(c) shows a photo of the Instron universal testing machine used forpeel and shear testing. The configuration of the testing machine and data sampling frequency is shownin Table 2.

Table 2. Testing protocol followed in our experiments

Test typeRamp slope

(mm/s)Sampling

frequency (ms)Joint

dimensionsPeel 1 10 8 mm diameter

Shear 0.3 10 8 mm diameter


8

(a) Specimen configu-ration for peel test

(b) Specimen configu-ration for shear test

(c) The Instron uni-versal testing ma-chine used in thisstudy

Figure 2. The configurations of peel and shear tests.

4. Results and discussion

In this section, results from the mechanical testing, including peel test and shear test, are presented,where different failure modes are identified. Moreover, different ML models are used to obtain theresponse surfaces, and their performance is compared with polynomial regression. Finally, the responsesurface models are used to identify the optimal parameters for a compound joint strength indicator.

4.1. Mechanical strength

0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80Weld Time [s]

400

500

600

700

800

900

1000

1100

1200

Max

imal Loa

d [N]

Maximal Load for Shear Test40 μm45 μm50 μm55 μm

(a) Shear strength

0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80Weld Time [s]

50

100

150

200

250

300

Maximal Load [N]

Maximal Load for Peel Test40 m45 m50 m55 m

(b) Peel strength

Figure 3. Mechanical strengths vs. welding time under different welding amplitudes.

We observe three different failure modes in both peel and shear tests, which are closely related tothe weld types. The formation of those failure modes can be ascribed to the reduction of base materialstrength and enhancement of joint strength. Due to the plastic deformation and longitudinal cyclicmotion of horn during welding, the base material is removed from the weld zone and flows to theperipheral area. Such material loss rises as the welding progresses, resulting in a decrease in base


9

material strength, meanwhile the ultrasonic bonding strength increases. Depending on the relativestrengths of base materials and joints, the weld types can be categorized in three types.

• Cold weld: The joint is not fully developed but the material base is hardly removed. In mechanicaltesting, only the joint is separated.• Good weld: The strength of joint and the base material are close or at the same level. In mechan-

ical testing, partial of joint and base materials are torn down.• Over weld: The base material at junction is weaker than the joint. In mechanical testing, only

base material adjacent to weld region is separated.

It should be noted that it is possible that an over weld in shear test can also be good or cold weld inpeel test, since the relative strength of base material to joint may vary in different loading types.

In this study, we select the maximum load as the joint strength in both shear and peel tests. Otherindicators such as energy absorbed during mechanical testing may also be used [13]. The outliers, dueto wrong welding configuration or inappropriate clamping position in mechanical testing, are removedusing a 1-sigma criterion. Fig. 3 shows the relationship between shear/peel strengths and welding timeunder different welding amplitude values.

The shear test result is shown in Fig. 3(a). For both 40 µm and 45 µm, the shear strength increasesas the welding time increases, while the slope of 40 µm is larger than that of 45 µm. The increasingtrend indicates that the joint forms progressively as the weld continues and the welding time increases.Additionally, 45 µm has a larger strength than 40 µm, but they converge to the same level when weldingtime reaches 0.65 s. On the other hand, the overall trends for 50 µm and 55 µm are similar, except thatfor 55 µm there is a drop in shear strength when welding time increases over 0.65. This indicates anincreased likelihood for over welds.

Table 3. Training and testing RMSEs of different methods. The last column refers to thesum of prediction errors for shear and peel strengths. The models with best performance ineach category are highlighted in bold.

Shear Strength RMSE [N] Peel Strength RMSE [N] Total TestingTraining Testing Training Testing RMSE

Spline 22±1 52±43 12±1 35±16 85GPR-RBF 29± 1 26±14 20± 8 24± 11 50GPR-RQ 7± 1 32± 11 2 ± 2 26± 12 58

SVR-RBF 30± 1 25± 13 20± 1 25± 15 50SVR-quad 30± 1 30± 9 21± 1 25± 13 55

Linear 53±1 37±13 41±1 34±15 73Quadratic 27±1 27±11 20±1 24±14 51

Cubic 26±1 28±12 19±1 26 ±15 54

It is observed from Fig. 3(b) that when the welding amplitude is low, the weld strength maintainsat a relatively high value. The trend for welding amplitude of 40 µm is most stable. This is becausethe amplitude of 40 µm is lower than the threshold to form a good weld. Even by increasing thewelding time, the central temperature and plastic deformation of weld region do not dramatically rise,hence almost no improvement in strength is observed. The trends for welding amplitudes of 40 µmand 45 µm both have a critical point, after which a significant drop in peel strength occurs, indicating


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the occurrence of over welds. The critical points are 0.75 s and 0.80 s for 45 µm and for 50 µm,respectively. When the welding amplitude reaches 55 µm, the weld strength is very low regardless ofwelding time. This is attributed to the fact that all welds are over welds, which lead to much lowerstrength.

Weld Amplitude [μm]

4045

5055

Weld T

ime [s]

0μ40μ5

0μ60μ7

0μ8

Max

Loa

d for S

hear Test [N]

500

550

600

650

700

750

(a) Response surface for the shear strength

40 45 50 55Weld Amplitude [μm]

0μ4

0μ5

0μ6

0μ7

0μ8

Weld Time [s]

540

565

590

615

640

665

690

715

740

765

Load

[N]

(b) Contour of the shear strength


4045

5055

Weld T

ime [s]

0μ40μ5

0μ60μ7

0μ8

Ma

Loa

d for P

eel T

est [N]

60

80

100

120

140

160

180

(c) Response surface for the peel strength


0μ4

0μ5

0μ6

0μ7

0μ8

Weld Time [s]

72μ0

79μ5

87μ0

94μ5

102μ0

109μ5

117μ0

124μ5

132μ0

Load

[N]

(d) Contour of the peel strength

Figure 4. Response surfaces obtained by GPR-RBF.

4.2. Joint strength prediction

To illustrate the performance of different models, the obtained response surfaces are evaluated onthe dataset using leave-two-out cross-validation, which is selected because of the small dataset wehave. In each round of cross-validation, the full dataset is randomly partitioned into a training anda test set, which include 32 and 2 configurations, respectively. Root-mean-square error (RMSE) isselected as the modeling performance indicator, because it is smoothly differentiable and minimizingsuch metric gives exactly the mean value [37]. RMSEs are averaged over all rounds to evaluate theprediction performance.

Table 3 shows the prediction accuracy of spline regression, GPR with RBF and RQ kernels, SVR


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with RBF and quadratic kernels, and polynomial regression with different orders. SVR-RBF has thesmallest testing error on shear strength, while SVR-quad and the quadratic function perform the bestaccuracy on peel strength. SVR-quad has smaller prediction uncertainty than the quadratic function.Though spline regression and SVR-RQ have small training errors for both shear and peel strengths,their testing errors are much larger, indicating an overfitting issue.


4045

5055

Weld T

ime [s]

0μ40μ5

0μ60μ7

0μ8

Max

Loa

d for S

hear Test [N]

500

550

600

650

700

750



0μ4

0μ5

0μ6

0μ7

0μ8

Weld Time [s]

568

588

608

628

648

668

688

708

728

748

Load

[N]



4045

5055

Weld T

ime [s]

0μ40μ5

0μ60μ7

0μ8

Ma

Loa

d for P

eel T

est [N]

6080

100

120

140

160

180



0μ4

0μ5

0μ6

0μ7

0μ8

Weld Time [s]

48

58

68

78

88

98

108

118

128

138

Load

[N]


Figure 5. Response surfaces obtained by SVR-RBF.

Fig. 4, 5, and 6 illustrate the response surfaces obtained by GPR-RBF, SVR-RBF, and quadraticregression, respectively. The topography and optimal parameters indicated by response surfaces aredifferent for different models. It is estimated that in GPR-RBF (Fig. 4b) and quadratic regression(Fig. 6b) the maximal shear strength occurs beyond the range of parameters selected in this study,whereas the maximum locates inside the parameter range in SVR-RBF (Fig. 5b). In contrast, all themaxima of peel strength are inside the parameter range (Fig. 4d, 5d and 6d). Interestingly, the responsesurfaces for shear and peel strengths are intrinsically different. The response surfaces for shear strengthall have two local maxima and the strengths of two peaks are comparable, as depicted in Fig. 4b, 5b,and 6b. Nevertheless, such phenomenon does not hold for the peel strength, for which there is only one


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4045

5055

Weld T

ime [s]

0μ40μ5

0μ60μ7

0μ8

Max

Loa

d for S

hear Test [N]

500

550

600

650

700

750



0μ4

0μ5

0μ6

0μ7

0μ8

Weld Time [s]

510

535

560

585

610

635

660

685

710

735

Load

[N]



4045

5055

Weld T

ime [s]

0μ40μ5

0μ60μ7

0μ8

Ma

Loa

d for P

eel T

est [N]

406080100120140160180



0μ4

0μ5

0μ6

0μ7

0μ8Weld Time [s]

37μ5

50μ0

62μ5

75μ0

87μ5

100μ0

112μ5

125μ0

137μ5

Load

[N]


Figure 6. Response surfaces obtained by quadratic regression.

peak (see Fig. 4d, 5d, and 6d). The peel strength decreases sharply when moving away from the globalmaximum. One possible explanation is that the damaged edges of over-weld joints in peel testingare easy to break due to the normal load exerted to the joint. Slight tearing of the joint surface willsignificantly deteriorate the joint strength. In contrast, the load in shear testing is applied in parallelwith the joint surface, so the shear strength is less sensitive to the tearing of the joint surface.

To determine the optimal weld parameters for both shear and peel strengths, the criterion of nor-malized sum is adopted, i.e., the goal is to maximize a compound strength indicator:

F =12

(Fpeel(x, y)Fpeel,max

+Fshear(x, y)Fshear,max

), (4.1)

where Fpeel and Fshear represent peel and shear strengths, respectively.The contour plot for F is illustrated in Fig. 7. The region in red color can be considered as good

weld region. It is seen that the welding time needed to form a good weld decreases approximately lin-early as the welding amplitude increases. When the welding amplitude exceeds the optimal value, the


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0μ4

0μ5

0μ6

0μ7

0μ8

Weld Time [s]

0μ726

0μ756

0μ786

0μ816

0μ846

0μ876

0μ906

0μ936

0μ966

0μ996

(a) GPR


0μ4

0μ5

0μ6

0μ7

0μ8

Weld Time [s]

0μ632

0μ672

0μ712

0μ752

0μ792

0μ832

0μ872

0μ912

0μ952

(b) SVR


0μ4

0μ5

0μ6

0μ7

0μ8

Weld Time [s]

0μ584

0μ624

0μ664

0μ704

0μ744

0μ784

0μ824

0μ864

0μ904

0μ944

(c) Quadratic regression (d) Ground truth

Figure 7. Contour plot of the sum of normalized maximal peel and shear load.

Table 4. Optimal parameters for the compound strength obtained by different methods.

Optimal ParametersWeld Amplitude Weld Time Optimal Value

GPR 43.0 0.80 0.996SVR 43.5 0.64 0.975

Quadratic 44.0 0.64 0.972Ground Truth 50.0 0.60 0.983


14

compound strength drops drastically no matter what the welding time is. This indicates the existenceof a threshold for welding amplitude, after which the over weld dominates. GPR-RBF, SVR-RBF, andquadratic regression give different global maxima, as shown by Table 4. Fig. 7d shows the groundtruth of the compound strength calculated from the experiment results. Though being low resolutionbecause of the limited number of experiments, Fig. 7d shows a ringlike pattern, which agrees withFig. 7a-c, and the threshold for welding amplitude is also consistent with all regression models. Com-pared with the ground truth (50 µm, 0.60 s), the outcomes by SVR-RBF and quadratic regression aremore accurate than that of GPR-RBF.

5. Conclusion

In this paper, we developed an ML-based RSM for two performance indicators in UMW–shearand peel strengths. The performance of spline regression, GPR, SVR, and polynomial regression wascompared using a dataset collected from UMW experiments where the values of welding amplitudeand welding time were varied. The trained response surface models were subsequently used to find theoptimal parameters for a compound strength indicator, which was defined as the average of normalizedshear and peel strengths. It was shown that ML methods hold good potential in modeling input-outputrelationships in UMW.

The experimental results indicate the existence of a threshold for the welding amplitude thresholdat which point over welds start to occur and dominate afterwards. A clear trend of welding strengthtransition in the peel strength is observed at 50 µm, implying the weld quality undergoes the transitionfrom cold weld to good weld, and to over weld.

A comparison between the response surfaces of shear and peel strengths leads to some interestingobservations. First, the shear strength stays at a much higher level than the peel strength. Second, theresponse surface of the shear strength presents two peaks with comparable strength values, but the peelstrength only has one global peak. Their distributions are also very different. The compound strengthindicator reaches its peak at modest values of both welding amplitude and welding time.

Future research can be conducted in the following directions. First, more welding parameters suchas clamping pressure and welding pressure will be introduced to create a larger dataset. It is expectedwhen more parameters are included, machine learning methods may gain a larger advantage due totheir advanced modeling capability. Second, process variability should be taken into account andappropriately modeled, because repeatable, reliable performance is of utmost importance to manufac-turers. Finally, other quality indices such as thermal and electrical conductivity should be considered,leading to an increased number of objectives for optimization.

Acknowledgments

We acknowledge support from the Advanced Manufacturing office (AMO) of the office of EnergyEfficiency and Renewable Energy (EERE) under the U.S. Department of Energy (DOE) through thecontract DE-EE0008312 and the National Science Foundation under Grant No. 1944345.


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Conflict of interest

The authors have no conflict of interest.

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multi-objective optimization of peel and shear strengths

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